Throughout this book we have depended in numerous ways on the idea of plausibility. It appears as “confidence value,” “symbolic plausibility value,” “degree of certainty,” “plausibilities of elementary hypotheses,” and the like.

Just what is this idea of plausibility? Can it be made precise? Is a plausibility a kind of likelihood, a kind of probability? What are the semantics? How much plausibility should be set for a given hypothesis under given circumstances? Is an objective standard possible? Does it even matter for a theory of intelligence, or for epistemology? Unfortunately, we cannot yet give definite answers to these questions.

The plausibility talk in this book is, in the first place, naturalistic. People mention plausibility when they describe their reasoning processes and when they write discussions of results in scientific papers. Plausibility talk seems to come naturally to us and is reflected in our ordinary language. Common usage allows both categorical judgments that something is or is not plausible, and graded comparative judgments, such as that one thing is “a lot” or “a little bit” more plausible than another. Throughout the work that we describe in this book we used our best judgments of when hypotheses were plausible, of how to evaluate evidence, and of how to weigh hypotheses. We were encouraged in our judgments by other people being likewise persuaded by appeals to “because it is the only plausible explanation,” and “more plausible than any other explanation for the data,” and the like. We were encouraged by the systems we built acting mostly according to our expectations, and producing correct answers.

Abduction machines – summary of progress

All six generations of abduction machines described in this book are attempts to answer the question of how to organize knowledge and control processing to make abductive problem solving computationally feasible. How can an intelligent agent form good composite explanatory hypotheses without getting lost in the large number of potentially applicable concepts and the numerical vastness of their combinations? What general strategies can be used? Furthermore, it is not enough simply to form the best explanation, which already appears to be difficult, but an agent needs to be reasonably sure that the explanation chosen is significantly better than alternative explanations, even though generating all possible explanations so that they can be compared is usually not feasible.

Thus it seems that we are in deep trouble. Logic demands that an explanation be compared with alternatives before it can be confidently accepted, but bounded computational resources make it impossible to generate all of the alternatives. So it seems, tragically, that knowledge is impossible! Yet we are saved after all by a clever trick; and that trick is implicit comparison. A hypothesis is compared with alternatives without explicitly generating them all. One way to do this, as we have seen, is by comparing parts of hypotheses. By comparing hypothesis-part h1 with hypothesis-part h2, all composite hypotheses containing h1 are implicitly compared with all composites containing h2. Another way to implicitly compare hypotheses is to rely on a hypothesis generator that generates hypotheses in approximate order of most plausible first.

Exponential sums are important tools in number theory for solving problems involving integers—and real numbers in general—that are often intractable by other means. Analogous sums can be considered in the framework of finite fields and turn out to be useful in various applications of finite fields.

A basic role in setting up exponential sums for finite fields is played by special group homomorphisms called characters. It is necessary to distinguish between two types of characters—namely, additive and multiplicative characters—depending on whether reference is made to the additive or the multiplicative group of the finite field. Exponential sums are formed by using the values of one or more characters and possibly combining them with weights or with other function values. If we only sum the values of a single character, we speak of a character sum.

In Section 1 we lay the foundation by first discussing characters of finite abelian groups and then specializing to finite fields. Explicit formulas for additive and multiplicative characters of finite fields can be given. Both types of characters satisfy important orthogonality relations.

Section 2 is devoted to Gaussian sums, which are arguably the most important types of exponential sums for finite fields as they govern the transition from the additive to the multiplicative structure and vice versa. They also appear in many other contexts in algebra and number theory. As an illustration of their usefulness in number theory, we present a proof of the law of quadratic reciprocity based on properties of Gaussian sums.

One of the major applications of finite fields is coding theory. This theory has its origin in a famous theorem of Shannon that guarantees the existence of codes that can transmit information at rates close to the capacity of a communication channel with an arbitrarily small probability of error. One purpose of algebraic coding theory—the theory of error-correcting and errordetecting codes—is to devise methods for the construction of such codes.

During the last two decades more and more abstract algebraic tools such as the theory of finite fields and the theory of polynomials over finite fields have influenced coding. In particular, the description of redundant codes by polynomials over Fq is a milestone in this development. The fact that one can use shift registers for coding and decoding establishes a connection with linear recurring sequences. In our discussion of algebraic coding theory we do not consider any of the problems of the implementation or technical realization of the codes. We restrict ourselves to the study of basic properties of block codes and the description of some interesting classes of block codes.

Section 1 contains some background on algebraic coding theory and discusses the important class of linear codes in which encoding is performed by a linear transformation. A particularly interesting type of linear code is a cyclic code—that is, a linear code invariant under cyclic shifts.

This chapter is of central importance since it contains various fundamental properties of finite fields and a description of methods for constructing finite fields.

The field of integers modulo a prime number is, of course, the most familiar example of a finite field, but many of its properties extend to arbitrary finite fields. The characterization of finite fields (see Section 1) shows that every finite field is of prime-power order and that, conversely, for every prime power there exists a finite field whose number of elements is exactly that prime power. Furthermore, finite fields with the same number of elements are isomorphic and may therefore be identified. The next two sections provide information on roots of irreducible polynomials, leading to an interpretation of finite fields as splitting fields of irreducible polynomials, and on traces, norms, and bases relative to field extensions.

Section 4 treats roots of unity from the viewpoint of general field theory, which will be needed occasionally in Section 6 as well as in Chapter 5. Section 5 presents different ways of representing the elements of a finite field. In Section 6 we give two proofs of the famous theorem of Wedderburn according to which every finite division ring is a field.

Many discussions in this chapter will be followed up, continued, and partly generalized in later chapters.

CHARACTERIZATION OF FINITE FIELDS

In the previous chapter we have already encountered a basic class of finite fields—that is, of fields with finitely many elements.

Sequences in finite fields whose terms depend in a simple manner on their predecessors are of importance for a variety of applications. Such sequences are easy to generate by recursive procedures, which is certainly an advantageous feature from the computational viewpoint, and they also tend to have useful structural properties. Of particular interest is the case where the terms depend linearly on a fixed number of predecessors, resulting in a so-called linear recurring sequence. These sequences are employed, for instance, in coding theory (see Chapter 8, Section 2), in cryptography (see Chapter 9, Section 2), and in several branches of electrical engineering. In these applications, the underlying field is often taken to be F2, but the theory can be developed quite generally for any finite field.

In Section 1 we show how to implement the generation of linear recurring sequences on special switching circuits called feedback shift registers. We discuss also some basic periodicity properties of such sequences. Section 2 introduces the concept of an impulse response sequence, which is of both practical and theoretical interest. Further relations to periodicity properties are found in this way, and also through the use of the so-called characteristic polynomial of a linear recurring sequence. Another application of the characteristic polynomial yields explicit formulas for the terms of a linear recurring sequence. Maximal period sequences are also defined in this section. These sequences will appear in various applications in later chapters.

This book is designed as a textbook edition of our monograph Finite Fields which appeared in 1983 as Volume 20 of the Encyclopedia of Mathematics and Its Applications. Several changes have been made in order to tailor the book to the needs of the student. The historical and bibliographical notes at the end of each chapter and the long bibliography have been omitted as they are mainly of interest to researchers. The reader who desires this type of information may consult the original edition. There are also changes in the text proper, with the present book having an even stronger emphasis on applications. The increasingly important role of finite fields in cryptology is reflected by a new chapter on this topic. There is now a separate chapter on algebraic coding theory containing material from the original edition together with a new section on Goppa codes. New material on pseudorandom sequences has also been added. On the other hand, topics in the original edition that are mainly of theoretical interest have been omitted. Thus, a large part of the material on exponential sums and the chapters on equations over finite fields and on permutation polynomials cannot be found in the present volume.

The theory of finite fields is a branch of modern algebra that has come to the fore in the last 50 years because of its diverse applications in combinatorics, coding theory, cryptology, and the mathematical study of switching circuits, among others.

The theory of polynomials over finite fields is important for investigating the algebraic structure of finite fields as well as for many applications. Above all, irreducible polynomials—the prime elements of the polynomial ring over a finite field—are indispensable for constructing finite fields and computing with the elements of a finite field.

Section 1 introduces the notion of the order of a polynomial. An important fact is the connection between minimal polynomials of primitive elements (so-called primitive polynomials) and polynomials of the highest possible order for a given degree. Results about irreducible polynomials going beyond those discussed in the previous chapters are presented in Section 2. The next section is devoted to constructive aspects of irreducibility and deals also with the problem of calculating the minimal polynomial of an element in an extension field.

Certain special types of polynomials are discussed in the last two sections. Linearized polynomials are singled out by the property that all the exponents occurring in them are powers of the characteristic. The remarkable theory of these polynomials enables us, in particular, to give an alternative proof of the normal basis theorem. Binomials and trinomials — that is, two-term and three-term polynomials—form another class of polynomials for which special results of considerable interest can be established. We remark that another useful collection of polynomials— namely, that of cyclotomic polynomials—was already considered in Chapter 2, Section 4, and that some additional information on cyclotomic polynomials is contained in Section 2 of the present chapter.

This book has been out of print for some time, but because of the continuing demand we want to make it available again. We have taken this opportunity to revise the book slightly. Historical and bibliographical notes have been added to each chapter, the bibliography has become more detailed, and the misprints in the original edition have of course been corrected. We would like to thank the staff at Cambridge University Press for accommodating our wishes so readily.

The subject of finite fields has undergone a spectacular development in the last 10 years. Great strides have been made, especially in the computational and algorithmic aspects of finite fields which are important in the rapidly developing areas of computer algebra and symbolic computation. The numerous applications of finite fields to combinatorics, cryptology, algebraic coding theory, pseudorandom number generation and electrical engineering, to name but a few, have provided a steady impetus for further research. Thus, the subject grows inexorably. Nevertheless, we hope that this book will continue to serve its purpose as an introduction for students, since it is devoted mainly to those parts that have a certain quality of timelessness, namely the classical theory and the standard applications of finite fields.

In this chapter we consider some aspects of cryptology that have received considerable attention over the last few years. Cryptology is concerned with the designing and the breaking of systems for the communication of secret information. Such systems are called cryptosystems or cipher systems or ciphers. The designing aspect is called cryptography, the breaking is referred to as cryptanalysis. The rapid development of computers, the electronic transmission of information, and the advent of electronic transfer of funds all contributed to the evolution of cryptology from a government monopoly that deals with military and diplomatic communications to a major concern of business. The concepts have changed from conventional (private-key) cryptosystems to public-key cryptosystems that provide privacy and authenticity in communication via transfer of messages. Cryptology as a science is in its infancy since it is still searching for appropriate criteria for security and measures of complexity of cryptosystems.

Conventional cryptosystems date back to the ancient Spartans and Romans. One elementary cipher, the Caesar cipher, was used by Julius Caesar and consists of a single key K = 3 such that a message M is transformed into M + 3 modulo 26, where the integers 0, 1, …, 25 represent the letters A, B, …, Z of the alphabet. An obvious generalization of this cipher leads to the substitution ciphers often named after de Vigenère, a French cryptographer of the 16th century.

Any nonconstant polynomial over a field can be expressed as a product of irreducible polynomials. In the case of finite fields, some reasonably efficient algorithms can be devised for the actual calculation of the irreducible factors of a given polynomial of positive degree.

The availability of feasible factorization algorithms for polynomials over finite fields is important for coding theory and for the study of linear recurrence relations in finite fields. Beyond the realm of finite fields, there are various computational problems in algebra and number theory that depend in one way or another on the factorization of polynomials over finite fields. We mention the factorization of polynomials over the ring of integers, the determination of the decomposition of rational primes in algebraic number fields, the calculation of the Galois group of an equation over the rationals, and the construction of field extensions.

We shall present several algorithms for the factorization of polynomials over finite fields. The decision on the choice of algorithm for a specific factorization problem usually depends on whether the underlying finite field is “small” or “large.” In Section 1 we describe those algorithms that are better adapted to “small” finite fields and in the next section those that work better for “large” finite fields. Some of these algorithms reduce the problem of factoring polynomials to that of finding the roots of certain other polynomials. Therefore, Section 3 is devoted to the discussion of the latter problem from the computational viewpoint.

Finite fields play a fundamental role in some of the most fascinating applications of modern algebra to the real world. These applications occur in the general area of data communication, a vital concern in our information society. Technological breakthroughs like space and satellite communications and mundane matters like guarding the privacy of information in data banks all depend in one way or another on the use of finite fields. Because of the importance of these applications to communication and information theory, we will present them in greater detail in the following chapters. Chapter 8 discusses applications of finite fields to coding theory, the science of reliable transmission of messages, and Chapter 9 deals with applications to cryptology, the art of enciphering and deciphering secret messages.

This chapter is devoted to applications of finite fields within mathematics. These applications are indeed numerous, so we can only offer a selection of possible topics. Section 1 contains some results on the use of finite fields in affine and projective geometry and illustrates in particular their role in the construction of projective planes with a finite number of points and lines. Section 2 on combinatorics demonstrates the variety of applications of finite fields to this subject and points out their usefulness in problems of design of statistical experiments.

In Section 3 we give the definition of a linear modular system and show how finite fields are involved in this theory.